Radiation Therapy With or Without Combination Chemotherapy or Pazopanib Before Surgery in Treating Patients With Newly Diagnosed Non-rhabdomyosarcoma Soft Tissue Sarcomas That Can Be Removed by Surgery

Inclusion Criteria: * Note: eligible patients must have a body surface area \>= 0.5 m\^2 AND be able to swallow whole tablets * Newly diagnosed and histopathologically confirmed, potentially resectable NRSTS of the extremity and trunk will be eligible for the chemotherapy or non-chemotherapy cohort based on: * Evidence of chemotherapy sensitivity of the histologic sarcoma subtype based on existing evidence from prior clinical trials * Sufficient risk of metastatic disease to warrant chemotherapy based on size and grade and * Medically deemed able or unable to undergo chemotherapy * Notes: an incisional biopsy or core biopsy is preferred; fine needle aspiration biopsy is not acceptable to establish the diagnosis * ELIGIBLE SITES: * Extremities: upper (including shoulder) and lower (including hip) * Trunk: body wall * INELIGIBLE SITES: Head and neck, visceral organs (with the exception of embryonal sarcoma of the liver), retroperitoneum, peritoneum, pelvis within the confines of the bony pelvis * ELIGIBILITY FOR CHEMOTHERAPY COHORT: * Stage T2a/b (\> 5 cm) and grade 2 or 3 AND * One of the following chemosensitive histologies as defined in the World Health Organization (WHO) classification of soft tissue tumors (with some evidence of good response to chemoradiation and of sufficient high risk of metastases, or clear evidence of metastases): * Unclassified soft tissue sarcomas that are too undifferentiated to be placed in a specific pathologic category in the WHO classification (often called "undifferentiated soft tissue sarcoma" or "soft tissue sarcoma not otherwise specified \[NOS\]") * Synovial sarcoma * Angiosarcoma of soft tissue * Adult fibrosarcoma * Mesenchymal (extraskeletal) chondrosarcoma * Leiomyosarcoma * Liposarcoma (excluding myxoid liposarcoma) * Undifferentiated pleomorphic sarcoma * Embryonal sarcoma of the liver * Patients meeting the above criteria (histology, size, and grade) with the EXCEPTION of histologies noted above may enroll on the chemotherapy cohort or the non-chemotherapy cohort at the discretion of the enrolling investigator; patients meeting these criteria with the EXCEPTION of histologies noted above but medically deemed unable to receive chemotherapy or who elect not to receive chemotherapy are eligible for the non-chemotherapy cohort * Patients with the following histologies are only eligible for the chemotherapy cohort and cannot enroll on the non-chemotherapy cohort: * Unclassified soft tissue sarcomas that are too undifferentiated to be placed in a specific pathologic category in the WHO classification (often called "undifferentiated soft tissue sarcoma" or "soft tissue sarcoma NOS") in patients \< 30 years of age * Synovial sarcoma * Embryonal sarcoma of the liver * ELIGIBILITY FOR NON-CHEMOTHERAPY COHORT: * Patients with any size of grade 2 or 3 of the following "intermediate (rarely metastasizing)" or "malignant" tumors, as defined in the WHO classification of soft tissue tumors for which we have consensus data of chemotherapy-resistance are eligible only for the non-chemotherapy cohort: * So-called fibrohistiocytic tumors - plexiform fibrohistiocytic tumor, giant cell tumor of soft tissues * Fibroblastic/myofibroblastic tumors - solitary fibrous tumor, malignant solitary fibrous tumor, inflammatory myofibroblastic tumor, low grade myofibroblastic sarcoma, myxoinflammatory fibroblastic sarcoma, atypical myxoinflammatory fibroblastic tumor, myxofibrosarcoma, low grade fibromyxoid sarcoma, sclerosing epithelioid fibrosarcoma * Tumors of uncertain differentiation - epithelioid sarcoma, alveolar soft part sarcoma, clear cell sarcoma of soft tissue, angiomatoid fibrous histiocytoma, ossifying fibromyxoid tumor, myoepithelioma, myoepithelial carcinoma, extraskeletal myxoid chondrosarcoma, neoplasms with perivascular epithelioid cell differentiation (PEComa), intimal sarcoma, atypical fibroxanthoma, mixed tumor NOS, phosphaturic mesenchymal tumor, malignant ossifying fibromyxoid tumor, malignant mixed tumor, malignant phosphaturic mesenchymal tumor * Chondro-osseous tumors - extraskeletal osteosarcoma * Pericytic (perivascular) tumors - malignant glomus tumor * Nerve sheath tumors - malignant peripheral nerve sheath tumor, malignant granular cell tumor, epithelioid malignant peripheral nerve sheath tumor, malignant Triton tumor * Undifferentiated sarcomas (with a specific pathologic category in the WHO classification) - undifferentiated round cell sarcoma, undifferentiated epithelioid sarcoma, undifferentiated spindle cell sarcoma * Patients meeting the criteria (histology, size, and grade) with the EXCEPTION of histologies noted above may enroll on the non-chemotherapy cohort at the discretion of the enrolling investigator; patients meeting these criteria with the EXCEPTION of histologies noted above but medically deemed unable to receive chemotherapy or who elect not to receive chemotherapy are eligible for the non-chemotherapy cohort; note that tumors arising in bone are NOT eligible for this study * Extent of disease: * Patients with non-metastatic and metastatic disease are eligible * Initially unresectable patients, with or without metastatic disease, are eligible as long as there is a commitment at enrollment to resect the primary tumor * Sufficient tissue and blood must be available to submit for required biology studies * Lansky performance status score \>= 70 for patients =\< 16 years of age * Karnofsky performance status score \>= 70 for patients \> 16 years of age * Absolute neutrophil count \>= 1500/uL; Note: no transfusions are permitted 7 days prior to laboratory studies to determine eligibility * Platelet count \>= 100,000/uL; Note: no transfusions are permitted 7 days prior to laboratory studies to determine eligibility * Hemoglobin \>= 8 g/dL for patients =\< 16 years of age; \>= 9 g/dL for patients \> 16 years of age; Note: no transfusions are permitted 7 days prior to laboratory studies to determine eligibility * Creatinine clearance or radioisotope glomerular filtration rate (GFR) \>= 70 mL/min/1.73 m\^2 or normal serum creatinine based on age/gender as follows: * 2 to \< 6 years; 0.8 mg/dL male; 0.8 mg/dL female * 6 to \< 10 years; 1 mg/dL male; 1 mg/dL female * 10 to \< 13 years; 1.2 mg/dL male; 1.2 mg/dL female * 13 to \< 16 years; 1.5 mg/dL male; 1.4 mg/dL female * \>= 16 years; 1.5 mg/dL male; 1.4 mg/dL female * Total bilirubin =\< 1.5 x upper limit of normal (ULN) for age * Serum glutamic oxaloacetic transaminase (SGOT) (aspartate aminotransferase \[AST\]) or serum glutamate pyruvate transaminase (SGPT) (alanine aminotransferase \[ALT\]) \< 2.5 x upper limit of normal (ULN) for age * Shortening fraction of \>= 27% by echocardiogram OR ejection fraction of \>= 50% by radionuclide angiogram * Corrected QT interval (QTc) \< 480 msec * No evidence of dyspnea at rest, no exercise intolerance, and a resting pulse oximetry reading \> 94% on room air if there is clinical indication for determination * Patients on low molecular weight heparin or Coumadin (with a stable international normalized ratio \[INR\]) are eligible * Patient must have a life expectancy of at least 3 months with appropriate therapy * All patients and/or their parents or legal guardians must sign a written informed consent * All institutional, Food and Drug Administration (FDA), and National Cancer Institute (NCI) requirements for human studies must be met Exclusion Criteria: * Patients with grade 1 NRSTS tumors of any size are not eligible * Patients with known central nervous system (CNS) metastases are not eligible; Note: brain imaging is not an eligibility requirement * Patients with evidence of active bleeding or bleeding diathesis will be excluded (Note: patients aged \> 17 years with excess of 2.5 mL of hemoptysis are not eligible) * Patients with gross total resection of the primary tumor prior to enrollment on ARST1321 are NOT eligible; patients who have experienced tumor recurrence after a gross total tumor resection are NOT eligible * Patients with uncontrolled hypertension are ineligible; uncontrolled hypertension is defined as follows: * Patients aged =\< 17 years: greater than 95th percentile systolic and diastolic blood pressure based on age and height which is not controlled by one anti-hypertensive medication * Patients aged \> 17 years: systolic blood pressure \>= 140 mmHg and/or diastolic blood pressure \>= 90 mmHg that is not controlled by one anti-hypertensive medication * Prior Therapy: * Patients must have had no prior anthracycline (e.g., doxorubicin, daunorubicin) or ifosfamide chemotherapy * Patients must have had no prior use of pazopanib or similar multi-targeted tyrosine kinase inhibitors (TKI) * Patients must have had no prior radiotherapy to tumor-involved sites * Note: patients previously treated for a non-NRSTS cancer are eligible provided they meet the prior therapy requirements; patients who have had chemotherapy or radiotherapy within 4 weeks (6 weeks for nitrosoureas or mitomycin C) prior to entering the study or those who have not recovered from adverse events due to agents administered more than 4 weeks earlier are excluded * Other types of invasive malignancy that are not disease free within 3 years except for non-melanoma skin cancer, lentigo maligna, any carcinoma-in-situ or prostate cancer with low risk factors * CYTOCHROME P450 3A4 (CYP3A4) substrates WITH narrow therapeutic indices: patients chronically receiving medications known to be metabolized by CYP3A4 and with narrow therapeutic indices within 7 days prior to study enrollment, including but not limited to pimozide, aripiprazole, triazolam, ergotamine and halofantrine are not eligible; Note: the use of fentanyl is permitted * CYP3A4 Inhibitors: patients chronically receiving drugs that are known potent CYP3A4 inhibitors within 7 days prior to study enrollment, including but not limited to itraconazole, clarithromycin, erythromycin many non-nucleoside reverse-transcriptase inhibitors (NNRTIs), diltiazem, verapamil, and grapefruit juice are not eligible * CYP3A4 Inducers: patients chronically receiving drugs that are known potent CYP3A4 inducers within 14 days prior to study enrollment, including but not limited to carbamazepine, phenobarbital, phenytoin, rifampin, and St. John's wort are not eligible (with the exception of glucocorticoids) * Certain medications that are associated with a risk for QTc prolongation and/or Torsades de Pointes, although not prohibited, should be avoided or replaced with medications that do not carry these risks, if possible * Subjects with any condition that may impair the ability to swallow or absorb oral medications/investigational product including: * Any lesion, whether induced by tumor, radiation or other conditions, which makes it difficult to swallow capsules or pills * Prior surgical procedures affecting absorption including, but not limited to major resection of stomach or small bowel * Active peptic ulcer disease * Malabsorption syndrome * Subjects with any condition that may increase the risk of gastrointestinal bleeding or gastrointestinal perforation, including: * Active peptic ulcer disease * Known intraluminal metastatic lesions * Inflammatory bowel disease (e.g., ulcerative colitis, Crohn's disease) or other gastrointestinal conditions which increase the risk of perforation * History of abdominal fistula, gastrointestinal perforation or intra-abdominal abscess within 28 days prior to beginning study treatment * Subjects with any of the following cardiovascular conditions within the past 6 months * Cerebrovascular accident (CVA) or transient ischemic attack (TIA) * Cardiac arrhythmia * Admission for unstable angina * Cardiac angioplasty or stenting * Coronary artery bypass graft surgery * Pulmonary embolism, untreated deep venous thrombosis (DVT) or DVT which has been treated with therapeutic anticoagulation for less than 6 weeks * Arterial thrombosis * Symptomatic peripheral vascular disease * Class III or IV heart failure as defined by the New York Heart Association (NYHA) functional classification system; a subject who has a history of class II heart failure and is asymptomatic on treatment may be considered eligible * History of serious or non-healing wound, ulcer, or bone fracture * Uncontrolled intercurrent illness including, but not limited to, ongoing or active infection, symptomatic congestive heart failure, unstable angina pectoris, cardiac arrhythmia, or psychiatric illness/social situations that would limit compliance with study requirements * Patients who are unable to swallow whole tablets are not eligible * Patients with a body surface area \< 0.5 m\^2 are not eligible * Human immunodeficiency virus (HIV)-positive subjects on combination antiretroviral therapy are ineligible because of the potential for pharmacokinetic interactions with pazopanib; in addition, these subjects are at increased risk of lethal infections when treated with marrow-suppressive therapy * Patients who are receiving any other investigational agent(s) * Pregnancy and breast feeding: * Female patients who are pregnant are ineligible due to risks of fetal and teratogenic adverse events as seen in animal/human studies * Lactating females are not eligible unless they have agreed not to breastfeed their infants during treatment and for a period of 1 month following completion of treatment * Female patients of childbearing potential are not eligible unless a negative pregnancy test result has been obtained * Unwillingness to use an effective contraceptive method for the duration of their study participation and for at least 1 month after treatment is completed if sexually active with reproductive potential